A Szemeredi-Trotter type theorem in $\mathbb{R}^4$

We show that under suitable non-degeneracy conditions, m points and n 2–dimensional algebraic surfaces in R satisfying certain “pseudoflat” requirements can have at most O ( mn + m + n ) incidences, provided that m ≤ n2− for any > 0 (where the implicit constant in the above bound depends on ), or m ≥ n. As a special case, we obtain the Szemerédi-Trotter theorem for 2–planes in R, again provided m ≤ n2− or m ≥ n. As a further special case we recover the Szemerédi-Trotter theorem for complex lines in C with no restrictions on m and n (this theorem was originally proved by Tóth using a different method). As a second special case, we obtain the Szemerédi-Trotter theorem for complex unit circles in C, which has applications to the complex unit distance problem. We obtain our results by combining the discrete polynomial ham sandwich theorem with the crossing number inequality.